Is the female doctor held responsible for the death of a child after giving him an injection by mistake, and what is the expiation for that, and can it be postponed or given to the poor or to hospitals?

We ask Allah to forgive you and overlook your sins. What you have done, misleading people into believing you are a doctor and treating them in fields outside your specialization, is considered a grave sin and necessitates compensation (daman). This is based on the saying of the Prophet (peace be upon him): "Whoever practices medicine without having been known to be a doctor before, he is liable." If someone practices medicine and does so ignorantly, they have deceived the patient, and compensation is due from them by the consensus of the scholars. Therefore, you are obligated to pay the blood money (diyah) for what you have committed, and it is legally incumbent upon your 'aqilah (male relatives on the father's side, who traditionally bear collective responsibility for certain financial obligations).

You must search for the rightful heirs and deliver the blood money to them. If this is not possible, you may give it as charity on their behalf to patients or other needy individuals. Once the rightful heirs are identified, they will have the choice to either approve the charity or take the blood money.

As for the expiation for killing, it is the emancipation of a slave. If you are unable to do so, then you must fast for two consecutive months. This expiation is required immediately for whoever is able to perform it.